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Bring a large pot of heavily salted water to a boil. Meanwhile, cook the sausage: In a sauté pan or skillet large enough to hold all the pasta, add the sausage and enough cold water to cover. Set over medium-high heat, then remove from the heat when the water hits a boil, about 8 to 10 minutes.
Transfer the sausage to a cutting board and cut into 1/2-inch coins. Dry out the pan and return it to the stove.
In the same pan, heat the olive oil over high until nearly smoking. Add the sausage and cook, flipping once, until dark brown on both sides, 5 to 7 minutes. Remove the sausage to a paper towel-lined plate, then reduce the heat to medium. Add the squash and a pinch of salt to the pan. Let cook, stirring briefly and scraping up any browned bits on the bottom of the pan, until browned, 5 to 7 minutes.
While the squash is browning, add pasta to boiling water, and cook to al dente according to package directions. Reserve 1 cup of the pasta cooking water, and drain the pasta.
When the squash is nicely browned, add the butter and sage and cook until the butter is golden, nutty smelling and foaming, just a minute or two, then immediately remove the pan from the heat and add back the sausage.
Add the pasta to the pan and mix with the brown-butter sauce (if the pasta has cooled off quite a bit, return the pan to low heat while you combine everything). Stir in the cheese, then add pasta water as needed to smooth the sauce. Adjust with salt and pepper, and serve with extra Parmesan, if you like.